Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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### Logical Equivalence and Truth Tables **Instruction:** Use a truth table to determine whether the two statements are equivalent: \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) and \(q \leftrightarrow \neg p\). **Truth Table:** | \(p\) | \(q\) | \(\neg p\) | \(\neg p \rightarrow q\) | \(q \rightarrow \neg p\) | \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) | \(q \leftrightarrow \neg p\) | |:----:|:----:|:---------:|:-----------------------:|:-----------------------:|:-----------------------------------------------------:|:-------------------------:| | T | T | F | T | F | F | F | | T | F | F | T | T | T | T | | F | T | T | T | T | T | T | | F | F | T | F | T | F | F | ### Analysis: - **Column \( \neg p \)**: Represents the negation of \( p \). - **Columns \( \neg p \rightarrow q \) and \( q \rightarrow \neg p \)**: Represents conditional statements. - **Column \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\)**: Represents the conjunction of the two conditional statements. - **Column \( q \leftrightarrow \neg p \)**: Represents the biconditional statement. ### Conclusion: The truth values in columns \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) and \( q \leftrightarrow \neg p \) are identical for all possible truth values of \( p \) and \( q \). Hence, the statements are equivalent. **Choose the correct answer below:** - \( \textcircled{O} \) The statements are equivalent. - \( \) The statements are not equivalent.
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Transcribed Image Text:### Logical Equivalence and Truth Tables **Instruction:** Use a truth table to determine whether the two statements are equivalent: \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) and \(q \leftrightarrow \neg p\). **Truth Table:** | \(p\) | \(q\) | \(\neg p\) | \(\neg p \rightarrow q\) | \(q \rightarrow \neg p\) | \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) | \(q \leftrightarrow \neg p\) | |:----:|:----:|:---------:|:-----------------------:|:-----------------------:|:-----------------------------------------------------:|:-------------------------:| | T | T | F | T | F | F | F | | T | F | F | T | T | T | T | | F | T | T | T | T | T | T | | F | F | T | F | T | F | F | ### Analysis: - **Column \( \neg p \)**: Represents the negation of \( p \). - **Columns \( \neg p \rightarrow q \) and \( q \rightarrow \neg p \)**: Represents conditional statements. - **Column \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\)**: Represents the conjunction of the two conditional statements. - **Column \( q \leftrightarrow \neg p \)**: Represents the biconditional statement. ### Conclusion: The truth values in columns \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) and \( q \leftrightarrow \neg p \) are identical for all possible truth values of \( p \) and \( q \). Hence, the statements are equivalent. **Choose the correct answer below:** - \( \textcircled{O} \) The statements are equivalent. - \( \) The statements are not equivalent.
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